J. Math. Biol. (1987) 25:553-570 Journal of

mathematical Biology

(~ Springer-Verlag 1987

Coexistence for systems governed by difference equations of Lotka-Volterra type

J. Hofbauer 1, V. Hutson 2 and W. Jansen 3

i Institut fiir Mathematik, Universit/it Wien, Strudlhofgasse 4, A-1090 Wien, Austria 2 Department of Applied Mathematics, The University, Sheffield Sl0 2TN, England 3 Akademie der Wissenschaften der DDR, Zentralinstitut fiir Kybernetik und Informationsprozesse, Kurstrasse 33, DDR-1086 Berlin, German Democratic Republic

Abstract. The question of the long term survival of species in models governed by Lotka-Volterra difference equations is considered. The criterion used is the biologically realistic one of permanence, that is populations with all initial values positive must eventually all become greater than some fixed positive number. We show that in spite of the complex dynamics associated even with the simplest of such systems, it is possible to obtain readily applicable criteria for permanence in a wide range of cases.

Key words: Difference equations - - Lotka-Volterra equations Permanence - - Discrete semi-dynamical systems

1. Introduction

Systems of difference equations have been much used in the modelling of the interactions of species with non-overlapping generations, see Hassell [3] for example. It is well known that even for one species the dynamics may be extremely complex, and it may be very difficult to predict the detailed asymptot ic behaviour. However, one of the most important questions from a biological point of view concerns the conditions under which long term survival of all the species is assured, and it is our aim to show here that notwithstanding the complex dynamics which may occur, it is often possible to give rather simple and complete answers to this question for an important class of models.

We consider here the system of difference equations

x xexp{r } ' aox j ( i = 1 , . . . , n) (1.1) j = l

on ~ , where x = ( X l , . . . , x , ) ' is the vector of populations at one generation, and x ' is the corresponding vector at the next generation. The ri are assumed to be constants and A = (ao) is a constant matrix. For closely spaced generations, these equations may be written in the form

xi( t + e) = xi( t ) exp{ e[ ri - ( A x ( t ) )i]},

554 J. Hofbauer et al.

and subtracting x~(t) from each side, dividing by e, and taking the limit as e ~ 0 we obtain the familiar system of Lotka-Volterra differential equations

~, = xi{r , - (Ax)~} (i = 1 , . . . , n). (1.2)

For n = 1, the map x' = x e r-ax has been studied in detail by May and Oster [19], and was shown to exhibit a similar scenario of chaotic behaviour for large r as its more popular but biologically less realistic counterpart x ' = r x ( 1 - x ) . Some two species cases have been considered, for example by Hassell [3], but higher dimensional difference systems of the type (1.1) have rarely been tackled in the literature. In this paper we want to show that (1.1) has been unjustly neglected, and is perhaps the most tractable discrete time model in Mathematical Ecology. We will point out that many results long known for (1.2) directly carry over to (1.1), in particular the crucial "averaging property", noticed originally by Volterra [23, pp. 173 et seq.], on the convergence of the time averages towards the equilibrium point (Lemma 2.4), which brings order from chaos. These remarks suggest that (1.1) is the natural analogue of (1.2) for discrete time and may therefore justifiably be termed a system of Lotka- Volterra difference equations.

The classical approach to questions of coexistence is through asymptotic stability or global asymptotic stability of a necessarily unique interior equilibrium point. However, from a biological point of view a more realistic criterion, that of permanence, appears to be that long term survival holds if and only if asymptotically (that is as the number of generations becomes very large), the species densities should become and remain greater than some fixed strictly positive quantity which is independent of the initial values if these are positive. This does not impose the restriction that the populations should settle down to fixed values, but on the other hand does not allow populations in the long run to remain near the boundary 0R+ of the nonnegative cone ~+ (corresponding to extinction of at least one species). In contrast, asymptotic stability and global asymptotic stability seem unrealistic in that they are too restrictive in requiring convergence to fixed values, while asymptotic stability is too weak in allowing orbits starting too far from the equilibrium point to converge to the boundary. We shall study the question of long term survival here from the point of view of permanence.

The system (1.1) is said to be permanent ~ if there is a compact (that is closed o n n

and bounded) set M in the interior R+ of R+, and thus with minimum distance from 0R+ greater than zero, such that for every initial value in ~}+ the orbits enter and remain within M. The system is weakly persistent if for any initial value in

o n

~+, lim su[~j_.~ xi ( j )> 0 (i = 1 , . . . , n), where x~(j) is the population of species i at generation j ; clearly permanence is a stronger property than weak persistence, although in certain circumstances the latter implies the former, see [2]. A property, for example permanence, is said to be robust if it is preserved under sufficiently small perturbations of the parameters in (1.1); it is clearly desirable that any condition in this area should be robust as it will not then be destroyed by small

1 The equivalent terms, cooperativeness, permanent coexistence, uniform persistence, and ecological stability have also been used in the literature

Difference equations and coexistence 555

changes in the specification of the system, and several of our results here are for robust permanence.

The idea of permanence is implicitly contained in Schuster et al. [20, 22] who studied a certain chemical network. Their ingenious technique has been simplified and adapted to yield a general method, based on the concept of an "average" Liapunov function, for establishing permanence by Hofbauer [4] and Hutson [11]. As shown by Jansen [16], this method is particularly effective for Lotka- Volterra equations because of the averaging property. There is now a considerable amount known concerning permanence in several different types of models, ranging from ordinary differential equations to partial differential equations (reaction-diffusion systems) and differential inclusions which allow for uncer- tainty in the specification of a system, see [1, 2, 7-15, 17, 24] for a wide range of applications. Permanence for general systems of difference equations has been studied by Hutson and Moran [ 14], a particular case being treated in [5]. However, our object here is to show that, because an analogous averaging property holds, for the special case of difference equations of Lotka-Volterra type (1.1) a great deal more can be said.

Sufficient conditions for permanence are obtained in Sect. 2; the mathematical technique used here is that of discrete semidynamical systems. Although the proofs are somewhat technical, the principal results, Theorem 2.5 and Corollary 2.6 are simple and easy to apply. Considerations of the limitation of the environ- ment dictate that for the model (1.1) to be realistic some condition of uniform boundedness must apply to the orbits; this question has only recently been resolved for Lotka-Volterra differential equations [6], and for difference equations there appears to be little information available. A discussion is given in Sect. 3. Necessary conditions for permanence based on degree theory are given in Sect. 4 and the results are combined in Sect. 5 to yield a complete characterization for robust permanence for three species problems. In one form these conditions take the form of a simple and attractive geometrical criterion, Theorem 5.2(c). Finally we give two applications in Sect. 6, the first pointing out the difference between conditions leading to asymptotic stability and permanence, the second dealing with perhaps one of the most difficult three species problem, the analogue of the May-Leonard system for three competing species.

2. Sufficient conditions for permanence

The principal aim of this section is to show that there is a set of algebraic conditions, straightforward to check in practice, at the (finite number of) equili- brium points on the boundary, which are enough to ensure permanence. This result, which is given in Theorem 2.5, also has a useful geometrical form, Corollary 2.6. A complementary result, Theorem 2.7, gives a similar criterion for showing that there is an attractor in 0 ~ , when of course the system is not permanent.

The theory of discrete semidynamical systems (abbreviated here to SDS) provides the basic mathematical tool, and some standard notation for these is first introduced. Let (X, d) be a locally compact metric space (which will here be a subset of ~_), and let T: X-> X be continuous. For k c 7/+ (the non-negative integers) put x . k = T k x , and note that x . 0 = x and x . (kl + k2) = (x. kl) �9 k2 for

556 J. Hofbauer et al.

kl, k2 ~ Z +. The semiorbit through x is the set

7+(x) = {y: y = x . k for some k c 7/+}.

The O-limit set, which essentially describes the asymptotic behaviour of the semiorbit, is defined to be

O(x) ={y: x" k i ~ y for some sequence (ki) -~ oc}.

For a subset Xo c X put

~*(Xo) = U ~+(x), O(Xo)= U O(x). x~Xo x~Xo

Xo is said to be forward invariant if TXo c Xo and strictly forward invariant if TXo c J~o- The set M is said to be absorbing for Xo if it is forward invariant and 7+(x) c~ M r 2~ for every x c Xo. As the terminology suggests, this means that every semiorbit starting in Xo is "sucked into" M from which it cannot then escape. Xo is a global attractor if limj_,oo d(x .j, Xo) = 0 for every x e X. The SDS is said to be dissipative if there is a bounded global attractor. The proof of the following lemma may be deduced directly from that of Hutson [11, Lemma 2.1] and is omitted. Lemma 2.1. Let U be open with compact closure, and suppose that V is open and forward invariant, where (J c V c X. Then/fy+(x) c~ U r O for every x c V, 7+(/J) is compact and absorbing for V.

Remark. For a dissipative SDS on R+ with global attractor )20 an application of the lemma to an open e-neighbourhood U of Xo shows that there is a compact absorbing set for ~ .

To frame the system (1.1) in this setting, rewrite it as

xl = xi exp{f(x)} (i = 1 . . . . , n), (2.1)

where

j = n

f ( x ) = r i - ~ a~xj. (2.2) j = l

This may be expressed as x ' = Tx where T:N~_~R+ is the continuous map with ith component

( Tx)~ = x~ exp{f(x)}. (2.3)

In dynamical systems terminology the condition of permanence may be rephrased as follows.

Definition. The system (2.1) is said to be permanent if and only if there exists a N+ of R+ which is contained in ~+ compact absorbing set M for the interior ~ " " ~ "

(and whose minimum distance from the boundary 0R~ is thus non-zero). It will be assumed throughout that the SDS on R~_ is dissipative, in which

case in view of the remark after Lemma 2.1 there is a compact absorbing set for R~_. We remark that dissipativity is equivalent to the possibly more familiar notion of ultimate or eventual uniform boundedness, that is the existence of a constant L such that

l i m s u p x ~ ( k ) < L ( i = 1 , . . . , n) k~oo

Difference equations and coexistence 557

for all x �9 N+. The question of dissipativity is itself of some interest, and is discussed for (2.1) in detail in Sect. 3. When this condition holds, the analysis may evidently be restricted to the compact absorbing set 7+({x: 0 ~< xi <~ L}). From here on we take this set to be X, which is therefore now a compact metric space, and let S=Xc~O~_. The condition of permanence may then be thought of intuitively as the requirement that S repels orbits in some strong uniform sense.

The role of the function P below will be discussed shortly after some simple algebraic identities have been derived. For any vector p = (p~, . . . , Pn) of strictly positive real numbers, define the continuous function P:X--> ~+ by setting

i ~ n

P(x) = [I xP', i ~ l

and note that P(x) = 0 if and only if x �9 S. Put

c~(m,x)=P(Tmx)/P(x) ( m � 9 1 4 9 (2.4)

Let

~)(X) = ~. Pifi(x) = ~ Pi aijxj . (2.5) i = 1 i ~ l 1

Then from (2.3) and (2.4),

c~(1, x) = exp{~b(x)}. (2.6)

Since ~b:X->N is continuous, (2.6) provides a continuous extension of a ( 1 , ' ) to X. Now for any kl, k2 �9 77+, x e X \S ,

c~(k, + k2, x) = P[x" (k,+k2)]/P(x)

P[x ' (k l+k2)] n(x" kl) P(x . k,) P(x)

--= a(k2, x ' k l )a (k l , x). (2.7)

In particular the relation c~(s,x)=a(1, Tx). a(1, x) provides a continuous extension of a ( 2 , . ) to X, and proceeding inductively, of a(m, . ) to X for all m �9 77 +. By repeated application of (2.7),

a ki, x = oL(k0, x) H a , x" Y~ ki (2.8) k i = 0 j = l i = 0

for any k o , . . . , kt �9 77+ and x �9 X, which implies

[j=m-I q~(x "j)~. (2.9) a(m, x ) : e x p [ j~o )

Define next

/3(x) = sup a(m, x), (2.10) m ~ l

and note that since P(x)> 0 if x e X \ S and P, T are continuous, fl(x) is finite for each x c X \ S . However, fl(x) may take the value +oo if x � 9 S.

558 J. Hofbauer et al.

We wish to show that orbits are repelled by S, and by a standard Liapunov function type argument we see that the condition a(1, x) = P ( T x ) / P ( x ) > 1 for all x in a neighbourhood of S, that is P is increasing along semiorbits, is enough to ensure this. However, this condition will not usually be satisfied for the type of system considered here, nor will it be easy to invent an alternative Liapunov function P. Our objective here is to show that the much weaker condit ion/3(x) > 1 for x ~ S, which requires only that P should increase over some (rather than every) section of the orbit, is sufficient for permanence. This justifies the term "average Liapunov function" for P. A preparatory technical lemma is needed.

Lemma 2.2. Let K a X be compact, and suppose that f l(x) > 1 for every x ~ K. Then there is a closed neighbourhood V of K such that if for some y c X we have ~ (y ) c V, then/3(y) = +oo, and y c S.

Proof The proof is divided into two stages. We first claim that there are a neighbourhood W of K, an /~> 0, and a /~< oo such that for any x c W there is a positive integer kx ~</c with a(kx, x) ~ 1 +/~ For each h > 0, k 6 Z + define the following subsets of X:

U(h, k) ={x: a(k, x ) > l+h} .

As a(k," ) is continuous, each U(h, k) is open, and since f l ( x ) > 1, from (2.10) these sets form an open cover of K. However, K is compact, so there is a finite subcover. Also U(hl, k) ~ U(h2, k) if hl<~ h2, so it follows that there are an /~> 0, integers k ~ , . . . , kj, and an open neighbourhood W of K with

K c W c UI----~ U(/~, k,).

This establishes the claim, for on each U(h, kj), a(kj, x)> 1+/~, and kj~ ]~= max k;.

To complete the proof, choose a closed neighbourhood V c W. I f for some y ~ X , ~ ( y ) c V, from the definition of J2-1imit set, for some ko, y" k e W for k ~> ko. From what was proved above, there is a sequence (kg) in 7/+ with 1 ~< k~ <~/~ such that for every i ~ > 1,

a(k,, y . (ko+" �9 �9 + ki-1)) >~ (1 +/T).

Hence from (2.8)

a ( i Z , i=0 ki' Y) >~ a( k~ y)( l + ~)t'

and it follows that ~(y) = 0% since a(ko, y) > 0. Finally, y c S, for as remarked previously, f i(y) is finite for each y ~ X \ S .

Our first sufficient condition for permanence is the following theorem. This is a special case of a result of Hutson and Moran [14], but its proof is an immediate consequence of Lemmas 2.1 and 2.2 which are needed later, and is included for completeness.

Theorem 2.3. Let the system (2.1) be dissipative. I f f l (x )> 1 for each x c S, per- manence holds.

D i f f e r e n c e e q u a t i o n s a n d c o e x i s t e n c e 559

Proof With the notation of Lemma 2.2, take K = S and N = X \ S , from that lemma 12 (x) c~ N # O , for otherwise x e S contrary to assumption. It then follows from Lemma 2.1 with U = N and V = X \ S that M = 7+(N -) is the required

R + . compact absorbing set in X \ S c ~ ~

Use of this theorem depends on estimates of fl at every point x e S. As shown in [14], it is in fact enough if f l ( x ) > 1 for each x 6 12(S). However, in view of the complexity of the 12-limit sets of difference equations even in low dimensions, it will often be difficult to check even this condition. Our next and principal objective is to show, by exploiting the averaging property as in [16], that it is sufficient if f i (x*)> 1 for the boundary equilibria x* only. At these points it is clear from (2.9) that f l(x*) = +oo if q~ (x*) > 0, so this criterion is readily checkable. We will also see in Sect. 4 that q~(x*) is a linear combination of the "external eigenvalues", which provides a relationship with the intuitive biological concept of "invasion parameter" .

Define the average population vector over m generations )7(m)= ( ) ~ l ( m ) , . . . , ~. (m)) by putting

k ~ m 1

Y,i(m)=m -1 Y~ ( x ' k ) i . k = 0

Then from the linearity of (2.5), (2.9) can be rewritten as

a (m, x) = exp{mq~(ff(m))}. (2.11)

Consider now a subsystem of (2.1) of order q obtained by setting n - q of the populations zero and rearranging the indices so that the non-zero populations correspond to i = 1, . . . , q.

Lemma 2.4. Assume that xi > 0 (1 ~ i <~ q ). Suppose that there are real numbers b > 0 and b', and a sequence (kj) -~ oo such that b < (x . kj)i < b' (1 <~ i ~< q, j /> 1). Then there are a subsequence, again denoted by (kj), and an equilibrium point x* such that

lim ff(kj) = x*. j ~oO

Proof The averages satisfy the same inequalities, so by compactness there is a convergent subsequence .~(kj). Then for 1 ~<i~ < q,

m=kj ky 1 log[(x" kj)i/xi] = k j - 1 Y, log[(x- m)i / (x" (m - 1))i]

rrl ~ k.

= k j 1 E~ f ( x ' ( m - 1 ) ) m = l

=f~(g(kj))

by the linearity of the f . The left-hand side tends to zero as j ~ oo, and hence the limit x* of the convergent subsequence .~(kj) satisfies f ( x * ) = 0. Thus x* is an equilibrium point of (2.1).

560 J. Hofbauer et al.

Theorem 2.5. Let the system (1.1) be dissipative. Assume that there are real numbers

P~ , �9 �9 �9 Pn > 0 such that

i = n

qS(x*) = Y, p i f ( x * ) > 0 (2.12) i=1

for each equilibrium point x* in OR+. Then (1.1) is permanent. I f (1.1) is robustly dissipative, robust permanence holds.

Proof Broadly this is carried out by induction on the dimension. Regarded as a subset of R+, OR+ is composed of n faces of dimension (n - 1) obtained by setting x ~ , . . . , x, zero in turn. Denote the intersection of each of these with X by F ("-1), and the set of all these faces by U F(n-1). Define F ( " -2 ) , . . . , F(~ {0} together with their interior and boundary similarly.

From (2.10)-(2.12), / 3 ( x * ) > l for each equilibrium point x * c S . From Theorem 2.3, it is enough to show that /3(x) > 1 for each x c S. Since the origin 0 is an equilibrium point,/3 (0) > 1. Suppose next that for some q with 0 ~< q ~< n - 2, /3(x) > 1 for all x e U F(q). Let x be an interior point of s o m e F (q+l). Then either g2(x) c U F(q), or g2(x) contains an interior point Xo, say, of F (q+~). In the first case, by the induction hypothesis and Lemma 2.2 with K = U F(q),/3(x) > 1. In the second case, there is a sequence (kj )~ oo with lim~_,oo x. kj = xo. By Lemma 2.4 the averages g(kj) converge to an equilibrium point x*, say, and it follows from (2.11) and (2.12) tha t /3(x) > 1. Thus /3(x)> 1 if x~ F (q+l), and the result follows by induction.

Note finally that for a sufficiently small change in the parameters of (1.1), the number of boundary equilibrium points cannot increase and their position is only slightly changed. Hence ~b remains positive at these points. Together with robust dissipativity this guarantees robust permanence.

Theorem 2.5 is our central result on sufficient conditions for permanence in Lotka-Volterra systems of difference equations. The principal point of this theorem is that it reduces the question of permanence to a (finite) algebraic problem, that of deciding when there exist n positive numbers Pi, such that the linear expressions ~b(x*) given by (2.5) are greater than zero at a finite number of equilibrium points. Some remarks concerning this follow.

(i) It can happen that there is a line of equilibrium points, but it has been shown in [16] that only the extreme points in R+ need be checked. Hence, as stated above, only a finite number of inequalities enter into the computation, so the problem is indeed purely algebraic. Furthermore, this algebraic problem is clearly identical with that for systems of Lotka-Volterra differential equations. A useful consequence of this is that the extensive results [7, 8] already available for the differential equation case may be used directly, and this will be done at several places in later sections.

It may also be useful to note that the problem is of linear programming type, so the powerful techniques available there may be exploited, see [16].

(ii) There is a simple geometrical condition for permanence for systems of Lotka-Volterra differential equations, see [7, 8] which in view of the remark above we may quote directly.

Difference equations and coexistence 561

Corollary 2,6, Let D denote the set of points where no species increases, that is

D = { x : r<- Ax},

where r = (r~, . . . , rn). Then if dissipativity holds, the system (1.1) is permanent if D is disjoint from the convex hull C of the boundaryfixed points.

(iii) We will see in Sect. 5 that, at least in low dimensions n ~ 3, the conditions in Theorem 2.5 or Corollary 2.6 in fact characterise the robust permanence of (1.1). For n >14 this remains an open question.

(iv) Whenever Theorem 2.5 characterises permanence there is a noteworthy consequence, which may be seen by setting a 0 = rico/Kj, and rewriting (1.1) in the form

, _ ( c o x j �9 x i - xi exp r i 1 -J j = l

Then permanence is completely independent of the basic growth rates ri and "carrying capacities", Ki, and is determined solely by the "communi ty matrix" (co). This holds under the restriction that changing the ri does not destroy the boundedness assumption, which is true for a large class of interaction schemes (see Lemma 3.3), but not for mutualists (see Lemma 3.2). On the other hand, as pointed out with some regret by Strobeck [25] for (1.2), the local stability of the interior fixed point may depend critically on the r~.

The following theorem, a generalization of a result in [1], is a partial converse to Theorem 2.5. It will be useful when necessary conditions for permanence are discussed in Sect. 5.

Theorem 2.7. Suppose (1.1) is dissipative. Assume that there is a subset M of S which is strictly forward invariant (for the SDS restricted to S), and that there are positive numbers P l , . . . , Pn > 0 such that oh(x* ) < 0 for every equilibrium point x* �9 M. Then M contains an attractor, that is there is a neighbourhood U of M in X such that Y2(x) c M for every x ~ U.

Proof This is based on arguments very similar to those used in proving Theorem 2.5 and is therefore only given in outline.

The result is first proved under the assumption that 6(x) < 1 for every x �9 M, where 6(x)=infm~>l a ( m , x ) . Since S2()~]r)c r / ~ c M, an argument similar to that of Lemma 2.2 (with the inequalities reversed) for the SDS restricted to. S shows that 6 ( x ) = 0 for every x �9 M. It follows that there are a neighbourhood W of M in X, an h > 0 and a / ~ < co such that if x �9 W there is a positive integer kx <~/~ with ce(kx, x ) < ~ l - h .

Let V be a neighbourhood of TM in X such that

V ~ S c M. (2.13)

Then U = Wc~ T - I ( v ) c ~ . . . n T-~(V) is a neighbourhood of 2~ in X. Since r~\ U is disjoint from S, by (2.13) there exists p > 0 such that P(x) > p for x ~ f ' \ U. Hence with I ( p ) = { x c X \ S : P ( x ) < p } , I ( p ) ~ V c U.

Let x � 9 U c W. Then by definition of W there is a k with l~<k~</~ such that a(k , x) <~ 1 - h. Hence P ( x . k) < P(x ) �9 (1 - h) <p , where x . k ~ I (p) .

562 J. Hofbauer et al.

Then x . k c Tk U c V and x . k ~ I ( p ) ~ V c U, from which it follows that x . k~ I ( p ) 71 U. By iteration we obtain a sequence (k~)--> co with k~+l- k~ <~/~ such that x . k~ ~ I ( p ) c~ U and P( x . k~) ~ O. Since P( x . k)<~ d P ( x . k~) holds for all k with k~<~k<~k~+l, where 6 = m a x { a ( m , x ) : O < ~ m < ~ k , x ~ X } , we conclude that P ( x . k ) ~ 0 and ~2(x)c M. Thus every orbit in I ( p ) c ~ U converges to M.

Finally, the averaging Lemma 2.4 implies again that 6(x*) < 1 at every equili- brium point x* in M is sufficient to ensure that 6(x) < 1 at every x c M.

3. Boundedness

The results throughout are based on the hypothesis that the system (1.1) is dissipative. It is clear from the finiteness of the universe that a hypothesis at least as strong as this is required for biological reality. However, the problem of deciding when the difference equations lead to dissipativity is not a simple one, and appears not to have received much attention in the literature. For the corresponding Lotka-Volterra differential equations (1.2) necessary and sufficient conditions are known. However, we shall show that whilst the analogous necessary conditions are still valid (Lemma 3.1), they are not sufficient (Lemma 3.2). It is curious to note that the sufficiency breaks down in what might be regarded as the most "cooperat ive" s y s t e m - - t h a t of a pair of muralists. Paradoxically the very fact that mutualists assist each other may lead to orbits which approach infinity, but which also come extremely close to both the axes, thus increasing the probability of extinction under external stresses. A sufficient condition (Lemma 3.3) is given which excludes such mutualist interactions.

We call the matrix A a B-matr ix if

Vx ~ ~+\{0} 3 i: x~ > 0 ancl (Ax)~ > 0. (3.1)

Biologically this property means that for the interaction matrix A in the Lotka- Volterra equations (1.2) or its discrete time analogue (1.1), at every possible state x of the system, at least one of the species, say i, "suffers" in the sense that its growth rate r~ is reduced to r~ - (Ax)g due to the interaction of the other species.

These matrices were studied in detail in [6]. In particular, it was shown there that (3.1) is equivalent to the fact that, for every choice of the ri, dissipativity holds for the Lotka-Volterra differential equations (1.2). The following lemma shows that for the difference equations, it is still a necessary condition.

Lemma 3.1. I f the sys tem (1.1) is dissipative and ri > 0 (or robustly dissipative and

the ri are arbitrary) then the interaction matr ix is a B-matrix.

Proo f Supbose A is not a B-matrix, then neither is its transpose A t (see [6]) and (3.1) gives us a vector p1>0, p # 0 with p i ( A p ) ~ O for all i. On the support I = {i: p~ > 0} of p we have

( a~p)~ = ( p ' a ) i <~ 0 (3.2)

for i ~ / , and so p" A x <~ 0 for all x with supp x c L We restrict our attention to this subsystem henceforth. It follows from (2.4)-(2.6) that P ( x ' ) > P ( x ) i f p . r > 0. I f p . r ~< 0 then we exploit the robustness assumption and by slightly decreasing the diagonal terms of A we can achieve strict inequality in (3.2). Then for large

Difference equations and coexistence 563

x with s u p p x = / , we get p . A x < p . r and again P ( x ' ) > P ( x ) . So the sets {x c R~+: P ( x ) >i a} are strictly forward invar iant for large a and all orbits in these sets go to infinity. This gives a contradict ion.

In contras t with the differential equat ion case, (3.1) is not sufficient to guaran- tee dissipativity of (1.1). This is shown by the fol lowing coun te rexample for two mutual is t ic species with interact ions mode l led by the equat ions

x ' = R x e x p ( - a x + by) (3.3)

y' = Ry exp(bx - ay),

with R, a, b > 0. The interact ion matr ix A is a B-mat r ix if

det A = a 2 - b 2 > 0, (3.4)

that is if the mutual is t ic effects are smaller than the intraspecific compet i t ion. N o w the line x = y is invariant , and there x ' = R x e x p [ ( b - a ) x ] , so for b > a (when by (3.4) A is not a B-matr ix) there are orbits converging to infinity, which is in accordance with L e m m a 3.1. However , the fol lowing l e m m a shows that in addit ion, for a > b u n b o u n d e d orbits exist if R is large enough.

Lemma 3.2. I f R > a / b > 1, then (3.3) has unbounded orbits.

Proof The idea is to construct orbits which start near the x-axis, that is with x large, y small, and which j u m p to points near the y-axis with x ' small and less than y, y ' large and greater than x. To be more precise we consider ho rn - shaped regions

BK, M = {(x, y) ~ ~2: x >! M, xy <~ K and y t> x e-bx}, (3.5)

where K > 0 is arbi t rary and M is any n u m b e r satisfying the inequalit ies

M >~ a - l , (3.6)

m / > 2(a - b) -1 log R, (3.7)

M >1 K a / l o g ( R b / a ) . (3.8)

I f B'K.M denotes the image of BK, M under the reflection (x, y)~--~(y, x ) then we claim that (3.3) maps BK, M into B'K, Ma/b. By iteration this proves that all orbits starting in BK, M are unbounded .

Let (x, y ) c BK, M. Then

(xy) ' = ( x y ) R 2 e -(a-b)(x+y) ~< xy <~ 1<2, (3.9)

since x >~ M and (3.7) holds. Moreove r

y' = y R e bx ay >1 x e-bX R e bx ay = R x e -ay.

Using y <~ K / x <~ K / M and (3.8) we obtain

y'>~ a x / b >t a M / b >1 1/b. (3.10)

Finally, since the funct ion y ~ y e -by is monoton ica l ly decreasing for y > 1/b we have

x ' / ( y ' e b Y ' ) > ~ x R e - a X + b y / ( a x e - O X / b ) = R b a l ebY>~l. (3.11)

(3.9), (3.10) and (3.11) imply (y', x ' ) c BK, Ma/V.

564 J. Hofbauer et al.

We conclude this section with a sufficient condition for boundedness. We call the interaction matrix A hierarchically ordered if there exists a rearrangement of the indices such that a~ ~ 0 whenever i ~<j, and additionally ai~ > 0 for every i. This means that positive interspecific effects (i.e. a o < 0) occur only from lower to higher levels. This excludes all types of mutalism, but allows for any kind of predator-prey or competitive interaction.

Lemma 3.3. I f A is hierarchically ordered then the difference equation (1.1) is dissipative (for every choice of the basic growth rates ri).

Proof Since a ~ > 0 for i~ j<-n , from (1.1) with i = 1 ,

x~ ~< xl exp(rl - allXl),

and it follows that x~ is bounded by the constant K1 = er:l/a11. Inserting this bound into (1.1) with i = 2, we find by the same argument that, after at most two generations, x2 is bounded by a constant /s Repeating the argument, we see that T" maps R+ into the compact set {x: 0 <~ x~ ~< K~}, which implies dissipativity.

Lemma 3.2 suggests that this result might be best possible in the sense that for every matrix A which is not hierarchically ordered and hence has what may be described as a "generalized mutualist cycle" (by which we mean a directed cycle of positive interspecific interactions), (1.1) will have unbounded orbits for large r~. This is in contrast with the situation for the differential equations where the result of Lemma 3.3 holds for all B-matrices.

4. Saturated equilibria and necessary conditions for persistence

Theorem 2.5 provides a sufficient condition for permanence. In an attempt to characterise permanence we now derive a set of necessary conditions, which are in fact valid even for weak persistence. Although these conditions are in general far from sufficient, we will see in Sect. 5 that they are strong enough to give characterisations if n <~ 3.

Consider an equilibrium point 2 of the difference equation (1.1) and let I = {i: xi > 0} be its support. The entries of the Jacobian matrix J(ff) of (1.1) at

are given by

OXl (t~ik_~,iaik) exp(r~--Y~ a~ffj), : i k ( =

[ 6ik -- g~a~k ( i C I) / \

Lo (ir i ~ k ) .

Hence J(ff) splits into two blocks: the index set I defines the "internal" block corresponding to the restriction of (1.1) to the species i ~ L The "external" block is a diagonal matrix whose entries are the external eigenvalues p~(ff) = expf~(~), where

f(Y~) = ri - 2 aox: (4.2) J

Difference equations and coexistence 565

are just the external eigenvalues of the Lotka-Volterra differential equations (1.2) at ~. Since pi(~) = limx_~ xJx i , pi(~) has the biological interpretation of a growth rate or "invasion parameter" of the (missing) species i at ~. At ff = 0, the p~ and f reduce to the basic growth rates R~ and r~ of each species.

The equilibrium ~ is called saturated (compare [8]) if all its external eigen- values p~(ff)<~ 1 (i~ I) or equivalently f(ff)<~ 0. If the system is at a saturated equilibrium ff then it is proof against the invasion of the missing species i ~/ . Note that an interior fixed point is always saturated since it has no external eigenvalues, while 0 is saturated only if it is semi-stable. The biological relevance of this concept relies on the following fact. If an orbit y+(x) with x ~ ~ R + converges, then its limit point is a saturated equilibrium. For our special system (1.1) the same also holds for the time averages if(m) (this being shown by an argument similar to that used in proving Lemma 2.4).

Conversely if ff is a strictly saturated equilibrium (i.e. pi(~) < 1 for all i~ I ) then the stable manifold of ff intersects the interior of R~_ and thus ff is the limit of interior orbits of (2.1).

In particular, permanence and even weak persistence are incompatible with the existence of a strictly saturated fixed point on the boundary. Moreover if weak persistence or permanence is a robust property of the system, saturated boundary fixed points may not occur either, as at least one of them could be made strictly saturated by a small perturbation. In other words, in a robustly weakly persistent system, at every boundary fixed point ~ at least one of the external eigenvalues p,(~) is larger than 1 or at least one of the f ( ~ ) is positive. In many cases (but not in general) this condition is also sufficient to guarantee weak persistence.

Permanence of (1.1), on the other hand, is guaranteed (and in many cases characterized) by Theorem 2.5 if a certain weighted sum Y. p ~ (~ ) of the logs of the external eigenvalues is positive at every boundary fixed point.

The internal block (i.e. the restriction to the index set I ) of the Jacobian J(ff) of (2.1), on the other hand, differs from the Jacobian matrix of the Lotka-Volterra equation (1.2) only by the identity matrix. Thus the internal eigenvalues of the difference equation are obtained by adding unity to the internal eigenvalues of the differential equation, in contrast to the connection (4.2) via the exponential function for the external eigenvalues.

Recall that the index of a vector field at a fixed point ~ is defined as the sign of the determinant of the Jacobian matrix of the vector field at that point (if is regular, that means that this determinant is non-zero). The above argument shows that at every regular fixed point ff the index with respect to the differential equation (1.2) is identical with its Lefschetz index for the map (2.1).

Lemma 4.1. Suppose that (2.1) is robustly dissipative (or that the interaction matrix A is a B-matrix). Then the difference equation (2.1) has at least one saturated fixed point. I f all saturated fixed points are regular then the sum of their indices equals ( -1)" .

Proof We have seen that the systems (1.1) and (1.2) have the same fixed points, the same saturated fixed points, and also their respective indices coincide. Now recall Lemma 3.1 and the fact [6] that when A is a B-matrix dissipativity holds

566 J. Hofbauer et al.

for the differential equations (1.2). The assertion of the lemma is then a con- sequence of the corresponding result for the differential equation proved in [7, 8].

We note that the above result is not restricted to (1.1) but holds for general maps on E+ with appropriate dissipativity assumptions.

Theorem 4.2. Suppose that (1.1) is robustly dissipative (or more simply that A is a B-matrix). Then the following conditions are necessary for (1.1) to be robustly permanent or robustly weakly persistent.

(i) (1.1) has an interior fixed point ~. (ii) det A > 0.

(iii) I f there exists a fixed point ~ with fik = 0 and Yi >~ 0 for i ~ k then the submatrix A (k~ with kth row and column deleted has positive determinant.

Proof As pointed out above, a robustly weakly persistent system cannot have a saturated fixed point on the boundary. Lemma 4.1 shows that there is at least one saturated fixed point, so an interior fixed point )7 > 0 must exist. This interior fixed point ~ is unique. Otherwise there would be a line of fixed points whose intersection with OR+ would still be a saturated fixed point. Uniqueness implies det A r 0 and hence regularity of ft. Then the second part of the lemma applies and shows that ind(~) = ( - 1 ) n or det A > 0 by (4.1). Finally (iii) is a consequence of the eigenvalue formula

fk() 7) = xi det A /de t A (k)

established in [7]. A partial converse of this is the following statement which can be proved in

the same way as was done for differential equations in [7]: if A is a P-matrix, that is all principal minors det(a~)i j~/> 0, and (1.1) has an interior fixed point then there is no saturated fixed point on the boundary and hence no interior orbit can converge to a point on the boundary.

5. The characterisation of permanence for three species

In this section we finally characterise (robust) permanence of (1.1) if the number of species is three or less. Surprisingly it turns out that the sufficient condition in terms of linear inequalities of Theorem 2.5, or the equivalent geometric condition of Corollary 2.6, actually characterise robust permanence. On the other hand the necessary conditions given in Theorem 4.2 characterise robust weak persistence and, up to one exceptional case, robust permanence as well. Thus, at least for n <~ 3? both the sufficient conditions obtained by use of an average Liapunov function in Sect. 2, and the necessary conditions obtained with index theory in Sect. 4 are good enough to give characterizations. This is noteworthy as difference equations, even with few species, are usually considered to be extremely hard and are rarely treated in the literature.

We start with the case of two species where the proof is an easy application of our previous results and is left to the reader, see also Example 6.1 below.

Theorem 5.1. I f (2.1) with n = 2 is robustly dissipative, then the following are equivalent.

(1) (2.1) is (robustly) permanent.

Difference equations and coexistence 567

(2) (2.1) is robustly weakly persistent. (3) The interior fixed point exists and det A = alia22- a12aza > O. (4) The interior fixed point lies outside the convex hull of the boundary fixed

points.

The study of three species systems is complicated by one exceptional case discovered by May and Leonard [18] for the differential equation, see Example 6.2. In this case interior orbits can converge to the boundary, but the S2-1imit sets are not contained in one boundary face. The system is then weakly persistent but not permanent.

More precisely we call the system (1.1) a system of May-Leonard type if r i > 0 ( i = 1 , 2 , 3 ) , a , > 0 (so the one-species fixed points Fl(r l /an , O, O), etc. exist), and the external eigenvalues pj(Fi) = exp A 0 at F~ have the following cyclic sign pattern: Al2, A23, A31/> 0 and A21, A32, AI~ ~< 0 (or this pattern reversed), where

Aij = rj - riaJ a,,

see Example 6.2. In this case we need to impose an additional condition to guarantee permanence.

(iv) I f (1.1) is of May-Leonard type then

~12/~23~31 + ~21/~32/~ 13 > 0. (5.1)

Theorem 5.2. Suppose (1.1) with n = 3 is robustly dissipative. Then the following holds (a) (1.1) is robustly weakly persistent if and only if properties (i), (ii) and (iii)

of Theorem 4.2 hold. (b) (1.1) is robustly permanent if and only if properties (i), (ii), (iii) and (iv) hold. (c) (1.1) is robustly permanent if and only if the two sets C and D of Corollary

2.6 are disjoint.

The corresponding result for the differential equation was established for two-prey one predator systems by Hutson and Vickers [9], for three competitors in [1], and for general three species systems by Hofbauer [7].

Proof Theorem 4.2 shows that (i), (ii) and (iii) are necessary conditions for robust weak persistence, respectively robust permanence. For the converse we use the result in [7]. The proof there shows that the algebraic conditions (i)-(iv) imply that the system of linear inequalities (2.12) has a positive solution p~ > 0. Thus Theorem 2.5 yields permanence.

We are left with showing that (iv) is necessary for robust permanence, and for this we proceed as in [1] or [7] using Theorem 2.7. We first establish the existence of a suitable strictly forward invariant set M. Since the origin 0 is a source, there is an open neighbourhood U of 0 such that the vector Tx points outwards if x c U\{0}. Now X \ U is compact, and so therefore is T ( X \ U). Further, clearly 0~ T ( X \ U ) . Thus there is a closed ball B centre 0 with B c U such that B ~ T ( X \ U ) = ~ . Since X is forward invariant, it follows that X \ B is strictly forward invariant, and so therefore is M = S ~ ( X \ B ) relative to S.

A simple calculation shows that the only boundary equilibrium points in M are the one species equilibrium points F~. It is also easy to see that ~t12~23/~31-t-

�9 ~21~32/~13<0 implies for all i the existence of p i > 0 such that ~jpjAij<0. I f equality occurs in (5.1) we can achieve this by a small perturbation of the

568 J. Hofbauer et al.

parameters using robustness. Thus P(x) = [I xf, is decreasing near the equilibrium points F~, and hence in the average near M, and by Theorem 2.7, M contains an attractor. Thus (1.1) is not permanent in this case. Since there is no saturated fixed point on the boundary, the system is nonetheless still weakly persistent.

6. Applications

As an illustration we consider two examples. The first shows that the criterion of asymptotic stability may be highly misleading.

Example 6.1. The difference equation analogue of the standard Lotka-Volterra model for a predator-prey interaction with x, y the respective populations may be expressed in the form

x'= x exp{r(1 - x ) - y } , (6.1)

y' = y exp{s(-1 +/3x)},

where r, s,/3 > 0. An elaboration of the argument in Lemma 3.3 shows that (6.1) is dissipative. Apart from the origin, there is only one boundary equilibrium /=1(1, 0), and there the external eigenvalue is exp A with A = s(/3 - 1). If A ~<0, that is/3 ~< 1, /=1 is saturated and it may be shown that y ~ 0 along all orbits. If A > 0 there is an interior equilibrium. The set C is the line segment of the x-axis from the origin to F1, and D is the intersection of {(x, y) ~ R2+: x ~</3-1} and the region to the right of the line r(1 - x ) - y = 0. If A > 0 these are disjoint, and by Corollary 2.6 the system is permanent. If intraspecific competition is introduced for the predator, (6.1) becomes robustly dissipative and Theorem 5.1 applies directly. The system is then robustly permanent iff there is an interior equilibrium, iff the boundary equilibrium F1 is not saturated.

When/3 > 1, the condition for asymptotic stability of the interior equilibrium is that both the following hold:

s ( / 3 - 1 ) < l ,

r [ 2 - s(/3 - 1)] < 4/3.

Thus here asymptotic stability is an unnecessarily restrictive condition for the long term survival of species. If r or s is large, it may be expected that the dynamics will be complicated, perhaps leading to chaos, but the system is still permanent.

Example 6.2. Consider the following difference equations for three competing species:

x~ = xl exp[r(1 - xl - c~x2 -/3x3)]

x~ = x2 exp[ r(1 - /3xl - x2 - c~x3)]

x~ = x3 exp[r(1 - aXl -/3x2 - x3)],

where 0 < a < 1 </3 and r > 0. These are the analogue of a set of Lotka-Volterra differential equations studied in [18, 21]. There are four boundary equilibria (0, 0, 0), (1, 0, 0), (0, 1, 0) and (0, 0, 1), and an interior equilibrium (1, 1, 1)/(1 + a +/3). The region D of decrease of all species is the intersection of those regions

Difference equations and coexistence 569

to the r ight o f each o f the p l a n e s 1 - Xl - a x 2 - / 3 x 3 = 0, 1 - /3Xl - x 2 - o/x 3 = 0 ,

1 - a x l - /3x2 - x3 = 0, a n d it is easy to check tha t this is d i s jo in t f rom the convex hu l l o f the b o u n d a r y e q u i l i b r i a i f a n d o n l y i f a + fl < 2. By T h e o r e m 5.2(c) this

c o n d i t i o n is neces sa ry a n d suff icient for r o b u s t p e r m a n e n c e . I f a +/3 > 2, in the d i f ferent ia l e q u a t i o n case there is a he t e roc l i n i c cycle j o i n i n g the b o u n d a r y equ i l i b r i a o n the pos i t ive c o o r d i n a t e axes w h i c h a t t racts all orbi ts (except for

those s ta r t ing on the l ine Xl = x 2 = x 3 ) , as s h o w n in [21]. F o r the d i f ference e q u a t i o n s , u n d e r the s a m e c o n d i t i o n s there is a n a t t rac to r in the b o u n d a r y , b u t o n gene ra l g r o u n d s this is l ike ly to be ex t r eme ly c o m p l e x i f r is large. I f o~ + 13 < 2 the s tab i l i ty of the i n t e r i o r e q u i l i b r i u m wil l be lost for large r, a n d the orbi ts will

be c o m p l e x p e r h a p s l e a d i n g to chaos , a l t h o u g h the sys tem is p e r m a n e n t . These e x a m p l e s s h o w tha t the c r i t e r ion for p e r m a n e n c e is r ead i ly checked

for th ree species by the use of T h e o r e m 5.2. F o r f o u r or m o r e species , whi ls t t h e o r e m s 2.5 a n d 2.6 still p ro v i d e a s imple way o f f ind ing suff icient c o n d i t i o n s

for p e r m a n e n c e , it is n o t k n o w n at the p r e s e n t t ime w h e t h e r these c o n d i t i o n s a r e a lso necessa ry .

Acknowledgements . The first two authors are grateful to the Mathematics Department, Northwestern University for its hospitality. The first author was supported in part by an Erwin Schr/Sdinger Stipendium of the Austrian FWF, the second by SERC grant number GR/D/84061.

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Received March 19/Revised August 12, 1987